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Up to now, jet kinematic studies of radio quasars have barely reached beyond the redshift range at $z>3.5$. This significantly limits our knowledge of high-redshift jets, which can provide key information for understanding the jet nature and the growth of the black holes in the early Universe. In this paper, we selected 9 radio-loud quasars at $z>3.5$ which display milliarcsec-scale jet morphology. We provided evidence on the source nature by presenting high-resolution very long baseline interferometry (VLBI) images of the sample at 8.4~GHz frequency and making spectral index maps. We also consider Gaia optical positions that are available for 7 out of the 9 quasars, for a better identification of the jet components within the radio structures. We find that 6 sources can be classified as core--jet blazars. The remaining 3 objects are more likely young, jetted radio sources, compact symmetric objects. By including multi-epoch archival VLBI data, we also obtained jet component proper motions of the sample and estimated the jet kinematic and geometric parameters (Doppler factor, Lorentz factor, viewing angle). Our results show that at $z>3.5$, the jet apparent transverse speeds do not exceed 20 times the speed of light ($c$). This is consistent with earlier high-redshift quasar measurements in the literature and the tendency derived from low-redshift blazars that fast jet speeds ($>40\,c$) only occur at low redshifts. The results from this paper contribute to the understanding of the cosmological evolution of radio AGN.